What does negative filling mean for Landau levels? I'm trying to read this paper, but I've never heard of Landau levels like $\nu=-1.$
What does it mean?
 A: First, let me say that I find the discussion in Hierarchy of fillings for the FQHE in monolayer graphene particularly clear on this topic, and Goerbig's RMP is also a good resource. The paper you mention focuses on the $n=0$ Landau level in graphene, so I will as well.
The basic idea is that there is a difference in how we label filling ratios in conventional semiconductor 2DEGs and in materials with Dirac-like spectra, such as graphene. In the 2DEG case $\nu_{semi}$ is measured relative to the empty lowest Landau level ($n=0$). Hence $\nu_{semi}\ge 0$ as you're probably familiar with.
In graphene it is natural to instead measure filling ratios relative to the neutrality point (the Dirac node). Due to particle-hole symmetry the $n=0$ Landau level is half-filled at zero energy. Hence $\nu=0$ in the graphene case corresponds to a half-filled Landau level, rather than an empty one as would be implied by $\nu_{semi}=0$, and negative filling ratios become possible.
The final ingredient to note is that the Landau levels in graphene have a four-fold degeneracy due to spin and valley symmetries. Applying a magnetic field breaks this degeneracy and you get four sublevels. In the case of the $n=0$ Landau level, two of the sublevels end up below the Dirac point, and two above it. The $n=0$ Landau level is then said to be empty at $\nu=-2$ and completely filled at $\nu=2$, and the graphene filling ratio can be related to the one in a 2DEG counting by $\nu = \nu_{semi}-2$.
